Neuromodulated Equilibrium Learning: Reward-Driven Learning from a Single Free Equilibrium

Equilibrium Propagation (EP) is a biologically motivated alternative to backpropagation but requires free and nudged equilibrium phases, raising questions about the biological plausibility of phase-separated learning. We introduce Neuromodulated Equilibrium Learning (NEL), a single-equilibrium reward-based framework inspired by Attention-Gated Brain Propagation and neuromodulatory signaling. After reaching a free equilibrium, NEL propagates a reward-dependent modulatory signal derived from the selected action through reciprocal feedback connections to drive local synaptic updates, eliminating both the reward-nudged equilibrium and contrastive update. We evaluate NEL on MNIST, Fashion-MNIST, and CIFAR-10 using multilayer perceptrons and convolutional networks, with symmetric three-phase REP and standard EP as multi-phase baselines. NEL approaches three-phase REP in shallow architectures, with gaps of 0.79 and 1.31 percentage points on MNIST and one-hidden-layer Fashion-MNIST, while larger gaps of 3.13 and 6.21 percentage points emerge in the deeper Fashion-MNIST and CIFAR-10 settings. A direct cost feedback control using the full supervised output error also remains below three-phase EP, indicating that sparse reward information alone does not explain the performance gap between single-equilibrium and multi-phase learning. These results show that effective reward-driven learning can arise from a single free equilibrium while suggesting that computations introduced by additional nudged relaxations become increasingly important in more challenging architectures.